Seiberg-Witten Theory and AGT Relation Tohru Eguchi We consider N = - - PowerPoint PPT Presentation

”Seiberg-Witten Theory and AGT Relation”

Tohru Eguchi We consider N = 2 supersymmetric gauge theories in 4- dimensions and study the case when the theory possesses the conformal invariance. Simplest example of a conformal invariant theory:

SU(2) gauge theory with Nf = 4 hypermultiplets

We may consider its generalizations A chain of SU(2) gauge theories with bifundamen- tals and fundamental at the ends: quiver gauge theories

As is well-known, such quiver theories are obtained using the brane construction as shown in the figure: One has n + 1 NS5 branes and a pair of D4 branes are sus-

pended between neighbouring NS5 branes giving rise to

SU(2)1 × SU(2)2 · · · × SU(2)n gauge symmetry. Two D4

branes at extreme left and right extend to x6 = ±∞ repre- senting fundamental hypermultiplets. In such a configuration each SUi(2) theory couples to Nf = 4 hypermultiplets and is conformally invariant. Thus there exists a set of marginal parameters in the theory

{τi = θi π + 8iπ g2

i

, i = 1, , n}

Uplifting this brane configuration to 11 dimensions

= ⇒ M theory picture with an M5 brane wrapping a Riemann

surface (cylinder) with punctures. Thus, conformal N = 2 theories

≈ an M5 brane wrapping a Riemann surface C with

a number of punctures. Number of parameters of Riemann surface Cg,n of genus g

with n punctures:

3g − 3 + n

This agrees with the number of gauge theory parameters {τi}. Hence one expects Gaiotto S-duality group of quiver gauge theory = mapping class group of Riemann surface Cg,n

Remarkable observation Alday,Gaiotto,Tachikawa AGT relation

⟨ ∏ Vmi(τi)⟩ = ∫ [da] |ZNek(τ; a; m, ϵi)|2

Liouville Nekrasov partition function correlation function

• f SU(2) gauge theory in Ω background

Liouville momentum

    

external line: mi

, ∆i = mi (Q − mi )

interbal line:

α = Q 2 + a

Background charges

Q = b + 1 b, c = 1 + 6Q2 ϵ1 = b, ϵ2 = b

! "# "$ "% "&

Nekrasov formula Sum over Yang tableau; Y = (λ1 ≥ λ2 ≥ · · · )

ZNek = ∑

(Y1,Y2)

q|⃗

Y |Zvector(⃗

a, ⃗ Y )Zantifund(⃗ a, ⃗ Y , m1) ×Zantifund(⃗ a, ⃗ Y , m2)Zfund(⃗ a, ⃗ Y , −m3)Zfund(⃗ a, ⃗ Y , −m4)

Here

Zvector(⃗ a, ⃗ Y ) = ∏

i,j=1,2

∏

s∈Yi

( aij − ϵ1LYj(s) + ϵ2(AYi(s) + 1) )−1 × ∏

t∈Yj

( aji + ϵ1LYj(t) − ϵ2(AYi(t) + 1) + ϵ+ )−1 Zfund(⃗ a, ⃗ Y , µ) = ∏

i=1,2

∏

s∈Yi

(ai + ϵ1(ℓ − 1) + ϵ2(m − 1) − µ + ϵ+) Zantifund(⃗ a, ⃗ Y , µ) = ∏

i=1,2

∏

s∈Yi

(ai + ϵ1(ℓ − 1) + ϵ2(m − 1) + µ) ϵ+ = ϵ1 + ϵ2, aij = ai − aj. LY (s) and AY (s) are leg and

arm length of the site s.

Nekrasov formula is obtained by summing over contributions from fixed points in the ADHM formula under gauge and Lorenz transformation (SO(4) = SU(2)L × SU(2)R ∈ (ϵ1, ϵ2)). First exact relationship between 4-dim CFT and 2-dim CFT. Higher rank generalization: Toda theories

♠ Attempts at direct proof

Fateev-Litvinov Detailed study of the algebraic structure of conformal block in Liouville theory , i.e. the recursion relation by Al.B.Zamolodchikov.

⟨Vα⟩ ≈ F∆

α (q),

F∆

α (q) =

∑ qmn Rn,m ∆ − ∆m,n F∆m,−n

α

(q)

and comparison with the sum over Yang tableaus of gauge theory side. Conformal block of 1-point function in Liouville theory on a torus = N = 4 gauge theory perturbed by the mass of the adjoint hypermultiplet (N = 2∗ theory)

♠ Exact Integration

Consider Liouville correlation function in free field represen- tation

⟨ ∏

a

eimaϕ(qa)

N

∏

i=1

∫ ebϕ(zi) dzi⟩

screening ops.

= ∏

a<b

(qa − qb)2mamb ∫ ∏

i,a

dzi (zi − qa)−2ibma ∏

i<j

(zi − zj)−2b2, ∑

i

ima + Nb = Q

Dotesnko-Fatteev integral

This is an integration of Selberg type.

IN(a, c, β) = ∫

N

∏

i=1

dxi ∏

i<j

(xi − xj)2β

N

∏

i=1

xa

i (1 − xi)c

=

N−1

∏

j=0

Γ(a + 1 + jβ)Γ(c + 1 + jβ)Γ(1 + (j + 1)β) Γ(a + c + 2 + (N + j − 1)β)Γ(1 + β)

Attempts at exact evaluation and comparison with conformal blocks. Morozov-Kironov-Shakirov, Itoyama-Oota· · ·

♠ Monodromy transformations

SW curve Σ :

x2 = ϕ2(z), ϕ2 has double poles ≈ m2

i

(z − zi)2 1 2πi

• Ai

xdz = ai, 1 2πi

• Bi

xdz = ai

D, ai D =

1 4πi ∂F ∂ai

In the semi-classical limit → 0,

ϵ1,2 << ai, mi Z ≈ exp ( −F (ai) 2 )

Liouville stress tensor T (z)

⟨T (z)Vm1(z1) · · · Vmn(zn)⟩ ≈ −1 2 ϕ2(z)⟨Vm1(z1) · · · Vmn(zn)⟩ ⇓ −m2

i

2

(z − zi)2 ≈ ∆i (z − zi)2

Degenerate field Consider a field Φ2,1(z) = e

−b 2 ϕ(z) which possesses a de-

generacy at level 2

∂2

z Φ2,1(z) = −b2 : T (z)Φ2,1(z) :

Correlation function with an extra insertion of Φ2,1

Z(ai; z) = ⟨Φ2,1(z)Vm1(z1) · · · Vmn(zn)⟩

In the semi-classical limit

Z(ai; z) ≈ exp ( −F (ai) 2 + bW (ai; z)

• + · · ·

)

One finds

(∂W )2 = ϕ2(z) = x(z)2

Hence

W±(z) = ± ∫ z

z∗ xdz

shift around A, B cycles gives

Z(ai; z + Aj) = exp(2πib

• aj)Z(ai; z)

Z(ai; z + Bj) = exp(2πib

• aj

D)Z(ai; z)

Similarly we may consider the process

• 1. Insert identity operator inside the Liouville correlator
• 2. Φ2,1 ⊗ Φ2,1 ≈ 1
• 3. Transport one of Φ2,1’s around A, B cycle
• 4. Pair annihilate two Φ’s into identity

= ⇒ L(γ)Fα = cos(πb(2α − Q) cos(πbQ) Fα

These processes give monodromy factors corresponding to the action of Wilson loop, ’t Hooft loop and surface operators. Alday-Gaiotto-Gukov-Tachikawa-Verlinde Drukker-Gomis-Okuda-Teschner

♠ Matrix Model

Dotsenko-Fatteev integral when b = i suggests a matrix model interpretation with an action

S = ∑

a

ma log(M − qa)

and {zi} are identified as matrix eigenvalues. Dijkgraaf-Vafa We find that this model in fact reproduces Seiberg-Witten the-

• ry (also for the asymptotically free cases Nf = 2, 3). But it

still has mysterious features. T.E.-Maruyoshi

Let us consider the simple case of 4 hypermultiplets with masses

m±, ˜ m±. Define m0 = 1 2(m+ − m−), m1 = 1 2( ˜ m+ − ˜ m−) m2 = 1 2(m+ + m−), m3 = 1 2( ˜ m+ + ˜ m−)

Condition:

∑

i

mi = 2gsN

M theory curve is given by

CM : (v − m+)(v − m−)z2 +c1(v2 + Mv − U)z + c1(v − ˜ m+)(v − ˜ m−) = 0

For convenience, set c1 = −(1 + q), c2 = q. By shifting v to eliminate the linear term and setting v = xz

CM : x2 = ( m2z2 + (1 + q)M

2 z + m3q

z(z − 1)(z − q) )2 +(m2

0 − m2 2)z2 − (1 + q)Uz + (m2 1 − m2 3)q

z2(z − 1)(z − q)

Seiberg-Witten differential behaves at a pole as

λSW = xdz 2πi ≈ m∗ z − z∗

Mass appears at residues. Pole at z = 0, z = ∞; residue ±m1, ±m0. Require pole at z = 1 with residue ±m2 and z = q with residue ±m3 =

⇒ M = −2q 1 + q(m2 + m3) ♣ UV and IR gauge coupling constant

Standard SW curve of Nf = 4 in massless case

CSW : y2 = 4x3 − g2ux2 − g3u3

Here

g2(ω1, q) = ( π ω1 )4 1 24 ( ϑ3(q)8 + ϑ2(q)8 + ϑ4(q)8) , g3(ω1, q) = ( π ω1 )6 1 432 ( ϑ4(q)4 − ϑ2(q)4) × ( 2ϑ3(q)8 + ϑ4(q)4ϑ2(q)4)

On the other hand M theory curve in the masssless limit is

given by

CM : x2 = − (1 + q)U z(z − 1)(z − q′)

Here U is related to u = trϕ2 as

U = Au

and we have used q′ in order to distinguish it from q of CSW . By comparing the periods we find

q′ = ϑ2(q)4 ϑ3(q)4, A = 1 ϑ2(q)4 + ϑ3(q)4

We regard q in SW curve as the gauge coupling in the infra- red regime q = qIR and q′ in M theory curve as the ultra- violet gauge coupling constant q′ = qUV . Relation

qUV = ϑ2(qIR)4 ϑ3(qIR)4

has been obtained by various authors. Grimm et al, Marshakov et al

♠ Matrix model and modular invariance

Equation of motion

∑ mi λI − qi + 2gs ∑

I̸=J

1 λI − λJ = 0

We have q1 = 0, q2 = 1, q3 = qUV . Eigenvalue distribution is as given in the figure.

Resolvent of the theory is defined by

Rm(z) = gsT r 1 z − M

and satisfies the loop equation

⟨Rm(z)⟩2 = −⟨Rm(z)⟩W ′(z) + f(z) 4 f(z) = 4gsT r ⟨ W ′(z) − W ′(M) z − M ⟩ =

3

∑

i=1

ci z − qi

Matrix model curve (spectral curve) is defined by the dis-

criminant of the loop eq.

Cspec.curve : x2 = W ′(z)2 + f(z) = (m1 z + m2 z − 1 + m3 z − q )2 + (m2

0 − ∑ i m2 i )z + qc1

z(z − 1)(z − q)

• Eq. of motion =

⇒ ∑

i

ci = 0

Residue at ∞ being ±m0 =

⇒ c2 + qc3 = m2

0 − (

∑ mi)2

Then

qc1 = (1 + q)m2

1 + (1 − q)m2 3 + 2qm1m2 − 2qm2m3

+2m1m3 − (1 + q)U = ⇒ CW = Cspec.curve

• Modular invariance

Consider the massless limit of spectral curve

x2 = − (1 + q)U z(z − 1)(z − q) = −

u θ4

3

z(z − 1)(z − q)

This is invariant under

I : (z, x) → (1 − z, x), q → 1 − q, u → −u, S II : (z, x) → (1 z, −z2x), q → 1 q, u → u, ST S

Recall q = θ4

2

θ4

3

.

Consider massive case. Under the S- and STS-transformations mass parameters are transformed into each other

I : (0, 1, q, ∞) → (1, 0, 1 − q, ∞), m1 ↔ m2 II : (0, 1, q, ∞) → (∞, 1, 1 q, 0), m0 ↔ m1

Under these transformations, the spectral curve should be in-

• variant. By imposing the conditions

x2(z; m0, m1, m2, m3; q) = x2(1 − z; m0, m2, m1, m3; 1 − q) x2(z; m0, m1, m2, m3; q) = 1 z4x2(1 z; m1, m0, m2, m3 : 1 q)

• ne can completely fix the mass dependence of the param-

eter U. Solution to the above conditions is given by

(1 + q)U = u ϑ4

3

− q(m2 + m3)2 + 1 + q 3  

3

∑

i=0

m2

i

 

• Asymptotically free theory with Nf = 3

precise relationship between u and T rϕ2

u = ⟨T rϕ2⟩ − 1 6(ϑ4

4 + ϑ4 3) 3

∑

i=0

m2

i .

Recall

m± = m2 ± m0, ˜ m± = m3 ± m1,

We take the limit

˜ m− → ∞, q → 0,

with

˜ m−q = Λ3

fixed

Matrix action reduces to

W (M) = ˜ m+ log M − Λ3 2M + m2 log(M − 1).

the spectral curve for Nf = 3 theory becomes

x2 = Λ2

3

4z4 − ˜ m+Λ3 z3(z − 1) − u − (m2 + 1

2 ˜

m+)Λ3 z2(z − 1) + m2 z(z − 1) + m2

2

z(z − 1)2 − m2Λ3 z2(z − 1).

Predicts the same free energy and discriminant as that of the

standard SW curve

x2 = z2(z − u) − 1 4Λ2

3(z − u)2

−1 4(m2

+ + m2 − + ˜

m2

+)Λ2 3(z − u) + m+m− ˜

m+Λ3z −1 4(m2

+m2 − + m2 − ˜

m2

+ + ˜

m2

+m2 +)Λ2 3

• Asymptotically free theory with Nf = 2

Matrix action:

W (M) = ˜ m+ log M − Λ2 2M − Λ2M 2

Spectral curve:

x2 = Λ2

2

4z4 + ˜ m+Λ2 z3 + u z2 + m+Λ2 z + Λ2

2

4

Computation of free energy

gsN1 = 1 4πi

• x(u)dz

(N1 denotes the filling fraction of the first cut ”1”). Derivative of free energy in Λ2 is given by

Λ2 ∂F ∂Λ2 = −Λ2 gs 2 ⟨ ∑

I

( 1 λI + λI ) ⟩ = 2u + Λ2

2 − m2 + − ˜

m2

+

On the other hand

4πi∂(gsN1) ∂u =

• C1

dz √ P4(z)

where

P4(z) = z4 + 4 ˜ m+ Λ2 z3 + 4u Λ2

2

z2 + 4m+ Λ2 z + 1

This is a complete elliptic integral and we can expand u in terms of a = 2gsN1 (we put m+ = ˜

m+ ≡ m for simplicity) u = a2 + m2 2a2Λ2

2 + a4 − 6m2a2 + 5m4

32a6 Λ4

2 + · · ·

Then by integrating over Λ2 we finally obtain

4Fm = 2(a2 − m2) log Λ2 + a2 + m2 2a2 Λ2 + a4 − 6a2m2 + 5m4 64a6 Λ4 + · · ·

This gives the same free energy as the standard SW curve

x2 = (z2 − 1 4Λ4

2)(z − u) + m+ ˜

m+Λ2

2z − 1

4(m2

+ + ˜

m2

+)Λ4 2

♠ Discussions

• 1. We want a much wider class of correspondences:

Liouville, Toda

= ⇒ WZW, cosets, parafermions etc. N = 2 Yang-Mills on R4 = ⇒ on ALE spaces, rational surfaces?

• 2. Want five-dimensional version of AGT. It is known that 5-

dimensional Nekrasov formula counts the number of holo- morphic curves in non-comapct CY manifolds (geometric engineering). 5-dim. AGT =

⇒ CFT acting on the space of

Gromov-Witten invariants.